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On regular anti-congruence in anti-ordered semigroups. (English) Zbl 1247.03133

Summary: For an anti-congruence \(q\) we say that it is a regular anti-congruence on the semigroup \((S,=, \neq, \cdot, \alpha)\) ordered under an anti-order \(\alpha\) if there exists an anti-order \(\theta\) on \(S/q\) such that the natural epimorphism is a reverse isotone homomorphism of semigroups. An anti-congruence \(q\) is regular if there exists a quasi-antiorder \(\sigma\) on \(S\) under \(\alpha\) such that \(q = \sigma\cup \sigma^{-1}\). Besides, for a regular anti-congruence \(q\) on \(S\), a construction of the maximal quasi-antiorder relation under \(\alpha\) with respect to \(q\) is shown.

MSC:

03F65 Other constructive mathematics
06F05 Ordered semigroups and monoids
20M99 Semigroups
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